Which of the following wave functions are eigenfunctions of...

Which of the following wave functions are eigenfunctions of the operator $d / d x ?$ If they are eigenfunctions, what is the eigenvalue? a. $a e^{-3 x}+b e^{-3 i x}$ b. $\sin ^{2} x$ c. e^{-i x}$ d. $\cos a x$ e. $e^{-i x^{2}}$

Transcript

00:01 Hello students, in this question we have been given some functions and we have to tell if those functions are eigenfunctions or not.

00:07 So let me tell you what an eigenfunction is.

00:09 Suppose we have a function represented by fx and we are applying an operator a cap on it.

00:15 Now if this operation gives back the same function multiplied with a constant value then this fx is known as an eigenfunction of this operator.

00:25 So this a is known as the eigenvalue of this function.

00:32 So our first function is a e raised to the power minus 3x plus b e raised to the power minus 3 iota x.

00:41 So this is the function and let's check it.

00:43 So we will apply the operator that is given.

00:46 So operator that is given to us is differential operator d by dx.

00:50 So this is what we will apply on this function.

00:54 So let's differentiate it.

00:56 So a will be constant derivative of e raised to the power minus 3x will be same and derivative of minus 3x will be equal to minus 3.

01:04 Now let's do the derivative of our second term.

01:07 B will be constant derivative of e raised to the power minus 3 iota x will be e raised to the power minus 3 iota x and now differentiation of this power will be equal to minus 3 iota.

01:19 Now we can take minus 3 common from here and we will finally get a e raised to the power minus 3x plus b iota.

01:31 So this iota will stay here e raised to the power minus 3 iota x.

01:35 Now this is the function that we have got and this is the constant value.

01:43 Now if we look at our initial functions this so these both functions are actually not the same because of this iota value.

01:52 So for being called as a eigenfunction both functions must be same.

01:57 So that is why the function given in option a is not an eigenfunction.

02:06 The second function that is given to us is sin square x.

02:09 So let's just differentiate it and sin square x can be written as 1 minus cos 2x divided by 2.

02:24 If you are wondering where this equation has come from we can calculate it by the formula of cos 2x.

02:32 So cos 2x is sin square x minus sin square x plus cos square x.

02:40 Now this cos square x can be replaced by 1 minus sin square x.

02:50 So cos 2x will be equal to 1 minus 2 sin square x.

02:56 We can rearrange this formula into this to calculate the value of our sin square x.

03:01 So sin square x will be 1 minus cos 2x divided by 2.

03:05 Now let's differentiate this function.

03:10 So this will be equal to d by dx of 1 by 2 minus d by dx of 1 by 2 cos 2x.

03:23 So derivative of constant will be 0 minus 1 by 2 will be taken out common.

03:30 Derivative of cos 2x is minus sin 2x and derivative of 2x is 2...

Question

Which of the following wave functions are eigenfunctions of the operator $d / d x ?$ If they are eigenfunctions, what is the eigenvalue? a. $a e^{-3 x}+b e^{-3 i x}$ b. $\sin ^{2} x$ c. e^{-i x}$ d. $\cos a x$ e. $e^{-i x^{2}}$

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